Berry Esseen Research Articles

A convolution inequality, yielding a sharper Berry–Esseen theorem for summands Zolotarev-close to normal

The classical Berry–Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment with a weak norm distance to normality, using Zolotarev’s ζ \zeta norms. We thus sharpen and simplify two results of Ul’yanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Kantorovich distances (often called Wasserstein distances, the L 1 ^1 norms of differences of distribution functions) of the corresponding factors, and also in the pair of the Lipschitz constants. Side results include a short introduction to ζ \zeta norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and Götze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.

Invariance Principle and Berry–Esseen Bound for Error Variance Estimator for Pth-Order Nonlinear Autoregressive Models

The invariance principle and Berry–Esseen bound for an error variance estimator based on the residuals are established by using a Taylor expansion and the classical invariance principle and Berry–Esseen bound for independent random variables. Some examples are given to illustrate their applications.

Uncertainty quantification and confidence intervals for naive rare-event estimators

Abstract We consider the estimation of rare-event probabilities using sample proportions output by naive Monte Carlo or collected data. Unlike using variance reduction techniques, this naive estimator does not have an a priori relative efficiency guarantee. On the other hand, due to the recent surge of sophisticated rare-event problems arising in safety evaluations of intelligent systems, efficiency-guaranteed variance reduction may face implementation challenges which, coupled with the availability of computation or data collection power, motivate the use of such a naive estimator. In this paper we study the uncertainty quantification, namely the construction, coverage validity, and tightness of confidence intervals, for rare-event probabilities using only sample proportions. In addition to the known normality, Wilson, and exact intervals, we investigate and compare them with two new intervals derived from Chernoff’s inequality and the Berry–Esseen theorem. Moreover, we generalize our results to the natural situation where sampling stops by reaching a target number of rare-event hits. Our findings show that the normality and Wilson intervals are not always valid, but they are close to the newly developed valid intervals in terms of half-width. In contrast, the exact interval is conservative, but safely guarantees the attainment of the nominal confidence level. Our new intervals, while being more conservative than the exact interval, provide useful insights into understanding the tightness of the considered intervals.

The Berry–Esseen bound in de Jong’s CLT

We prove a Berry–Esseen bound in de Jong’s classical CLT for normalized, completely degenerate U-statistics, which says that the convergence of the fourth moment sequence to three and a Lindeberg–Feller type negligibility condition are sufficient for asymptotic normality. Our bound is of the same optimal order as the bound on the Wasserstein distance to normality that has recently been proved by Döbler and Peccati (2017).

The improved correlation coefficient of Chatterjee

Chatterjee's new coefficient proposed by Chatterjee [2021, ‘A New Coefficient of Correlation’, Journal of the American Statistical Association, 116(536), 2009–2022.] is used to measure the degree of dependence between two scalars by rank correlation. However, the independence test based on Chatterjee's rank correlation may lose power since it only considers the distance between the nearest neighbours and ignores the other neighbours. In this paper, we propose an improvement to Chatterjee's new coefficient by incorporating the inverse distance-weighting, and further obtain the asymptotic normality of the improved coefficient and the Berry–Esseen bound under the null hypothesis. The proposed method is evaluated on the simulated as well as the real data on Yeast Gene Expression. The results show that the proposed method is superior to Chatterjee's new coefficient under various alternative hypotheses.

Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

Uniform Cramér moderate deviations and Berry–Esseen bounds for superadditive bisexual branching processes in random environments

On the rate of normal approximation for Poisson continuum percolation

It is known that the cardinality of the largest cluster of a percolating Poisson process restricted to a large finite box is asymptotically normal. In this note, we establish a rate of convergence for the statement. As each point in the largest cluster is determined by points as far as the diameter of the box, known results in the literature of normal approximation for Poisson functionals appear to be inapplicable. To disentangle the long-range dependence of the largest cluster, we use the fact that the second largest cluster has comparatively shorter range of dependence to restrict the range of dependence, apply a recent result of Chen et al. (2021) to obtain a Berry–Esseen type bound for the normal approximation of the number of points belonging to clusters that have a restricted range of dependence, and then estimate the gap between this quantity and the cardinality of the largest cluster.

An exponential nonuniform Berry–Esseen bound of the maximum likelihood estimator in a Jacobi process

AbstractWe establish the exponential nonuniform Berry–Esseen bound for the maximum likelihood estimator of unknown drift parameter in an ultraspherical Jacobi process using the change of measure method and precise asymptotic analysis techniques. As applications, the optimal uniform Berry–Esseen bound and optimal Cramér-type moderate deviation for the corresponding maximum likelihood estimator are obtained.

Statistical inference for scale mixture models via Mellin transform approach

This paper deals with statistical inference for the scale mixture models. We study an estimation approach based on the Mellin–Stieltjes transform that can be applied to both discrete and absolute continuous mixing distributions. The accuracy of the corresponding estimate is analysed in terms of its expected pointwise error. As an important technical result, we prove the analogue of the Berry–Esseen inequality for the Mellin transforms. The proposed statistical approach is illustrated by numerical examples.

Asymptotic normality of Nadaraya–Waton kernel regression estimation for mixing high-frequency data

High-frequency data is widely used and studied in many fields, especially in the econometrics and statistics. In this paper, the asymptotic normality of Nadaraya–Waton (NW) kernel regression estimator under ρ-mixing high-frequency data is studied. We first derive some moment inequalities for ρ-mixing high-frequency data, and then use them to study the asymptotic normality of NW kernel regression estimator, and give Berry–Esseen upper bounds. The numerical simulations report that the kernel regression estimator of high-frequency data has good asymptotic normality. Our empirical analysis is to fit the correlation between the five minute interval price increment and the corresponding trading volume for the Shanghai Stock Exchange Index, Entrepreneurship Index and Real Estate Index. These kernel regression curves better reflect people's investment behaviour.

Probabilistic limit theorems induced by the zeros of polynomials

AbstractSequences of discrete random variables are studied whose probability generating functions are zero‐free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to Berry–Esseen bounds, moderate deviation results, concentration inequalities, and mod‐Gaussian convergence. In addition, an alternate proof of the cumulant bound with improved constants for a class of polynomials all of whose roots lie on the unit circle is provided. A variety of examples is discussed in detail.

The Berry–Esseen theorem for circular β-ensemble

The Berry–Esseen theorem for circular β-ensemble

Refinements of Berry–Esseen Inequalities in Terms of Lyapunov Coefficients

Refinements of Berry–Esseen Inequalities in Terms of Lyapunov Coefficients

Vector-valued statistics of binomial processes: Berry–Esseen bounds in the convex distance

Vector-valued statistics of binomial processes: Berry–Esseen bounds in the convex distance

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

AbstractGiven a Gromov‐hyperbolic group endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.

A Berry–Esseen theorem for sample quantiles under martingale difference sequences

In this paper, we establish the uniformly asymptotic normality for sample quantiles based on martingale difference sequences under some suitable conditions. We obtain the rate of normality approximation of O ( n − 1 / 4 log ⁡ n ) by using some classical methods such as Bernstein type inequality, and so on. Finally, we verify asymptotic normality for the fixed quantile of the martingale difference sequences and present some numerical simulations to demonstrate the finite sample performances of the theoretical results.

Hölder's inequality and its reverse—A probabilistic point of view

AbstractIn this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder inequality for random vectors in . We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on balls and spheres. We also provide a Berry–Esseen–type result and prove a large and a moderate deviation principle for the suitably normalized Hölder ratio.

Local limit theorems for collective risk models

A probability model known as the collective risk model is used to describe the total claim from a portfolio of insurance contracts. It is essential to non-life insurance. Let N represent the number of claims and X1,X2,… represent the amount of loss in each claim where Xj′s are independent and identically distributed. For the collective risk model, the total claim is given by SN=X1+X2+⋯+XN. Local limit theorems estimate the probability at a particular point P(SN=k). In this paper, we provide local limit theorems for SN, where N is a random variable with a binomial, Poisson and negative binomial distribution. Our results give a better rate of convergence than Berry–Esseen’s Theorem. Explicit constants of the error bounds are also given.

Anticoncentration and Berry–Esseen bounds for random tensors

We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions $$\begin{aligned} \sum _{i\in [n]^d} \theta _i X_i \end{aligned}$$ of random tensors $$\varvec{X}=\langle X_i:i\in [n]^d\rangle $$ which are symmetric and exchangeable, and whose entries have bounded third moment and vanish on diagonal indices. These estimates are expressed in terms of intrinsic (and easily computable) parameters associated with the random tensor $$\varvec{X}$$ and the given coefficients $$\langle \theta _i:i\in [n]^d\rangle $$ , and they are optimal in various regimes. The key ingredient—which is of independent interest—is a combinatorial CLT for high-dimensional tensors which provides quantitative non-asymptotic normality under suitable conditions, of statistics of the form $$\begin{aligned} \sum _{(i_1,\dots ,i_d)\in [n]^d} \varvec{\zeta }\big (i_1,\dots ,i_d,\pi (i_1),\dots ,\pi (i_d)\big ) \end{aligned}$$ where $$\varvec{\zeta }:[n]^d\times [n]^d\rightarrow \mathbb {R}$$ is a deterministic real tensor, and $$\pi $$ is a random permutation uniformly distributed on the symmetric group $$\mathbb {S}_n$$ . Our results extend, in any dimension d, classical work of Bolthausen who covered the one-dimensional case, and more recent work of Barbour/Chen who treated the two-dimensional case.

Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process

AbstractLet $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.